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Well-posedness, long-time behavior, and discretization of some models of nonlinear acoustics in velocity-enthalpy formulation

Herbert Egger, Marvin Fritz

TL;DR

By recasting nonlinear acoustics models in velocity-enthalpy form, the paper uncovers a port-Hamiltonian structure that yields passive energy behavior. It proves local and global well-posedness for small initial data and exponential decay to equilibrium, based on energy $H(v,h)$ and a stronger energy $E(v,h)$, and a Haraux–Lagnese inequality. A structure-preserving discretization via mixed finite elements and an implicit time-stepping scheme demonstrates discrete energy dissipation consistent with the continuum, with numerical tests comparing Westervelt, Kuznetsov, and Rasmussen. The results provide a rigorous plus practical framework for stable long-time simulations of nonlinear acoustics and motivate further discretization error analysis and dispersive extensions.

Abstract

We study a class of models for nonlinear acoustics, including the well-known Westervelt and Kuznetsov equations, as well as a model of Rasmussen that can be seen as a thermodynamically consistent modification of the latter. Using linearization, energy estimates, and fixed-point arguments, we establish the existence and uniqueness of solutions that, for sufficiently small data, are global in time and converge exponentially fast to equilibrium. In contrast to previous work, our analysis is based on a velocity-enthalpy formulation of the problem, whose weak form reveals the underlying port-Hamiltonian structure. Moreover, the weak form of the problem is particularly well-suited for a structure-preserving discretization. This is demonstrated in numerical tests, which also highlight typical characteristics of the models under consideration.

Well-posedness, long-time behavior, and discretization of some models of nonlinear acoustics in velocity-enthalpy formulation

TL;DR

By recasting nonlinear acoustics models in velocity-enthalpy form, the paper uncovers a port-Hamiltonian structure that yields passive energy behavior. It proves local and global well-posedness for small initial data and exponential decay to equilibrium, based on energy and a stronger energy , and a Haraux–Lagnese inequality. A structure-preserving discretization via mixed finite elements and an implicit time-stepping scheme demonstrates discrete energy dissipation consistent with the continuum, with numerical tests comparing Westervelt, Kuznetsov, and Rasmussen. The results provide a rigorous plus practical framework for stable long-time simulations of nonlinear acoustics and motivate further discretization error analysis and dispersive extensions.

Abstract

We study a class of models for nonlinear acoustics, including the well-known Westervelt and Kuznetsov equations, as well as a model of Rasmussen that can be seen as a thermodynamically consistent modification of the latter. Using linearization, energy estimates, and fixed-point arguments, we establish the existence and uniqueness of solutions that, for sufficiently small data, are global in time and converge exponentially fast to equilibrium. In contrast to previous work, our analysis is based on a velocity-enthalpy formulation of the problem, whose weak form reveals the underlying port-Hamiltonian structure. Moreover, the weak form of the problem is particularly well-suited for a structure-preserving discretization. This is demonstrated in numerical tests, which also highlight typical characteristics of the models under consideration.
Paper Structure (11 sections, 6 theorems, 58 equations, 3 figures)

This paper contains 11 sections, 6 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. A linearized problem
  4. Local solvability
  5. Global solutions and decay estimate
  6. Numerical illustration
  7. Structure-preserving discretization
  8. Fully discrete scheme
  9. A one-dimensional example
  10. A two-dimensional example
  11. Conclusion and outlook

Key Result

proposition 1

Let $(v,h)$ be a sufficiently smooth solution of eq:sys1--eq:sys4 on $[0,T]$ for some $T>0$. Then for all $w \in L^2(\Omega)^n$ and $q \in H_0^1(\Omega)$, and all $0 \le t \le T$. Moreover, where $\mathcal{H}(v,h) := \int_\Omega \tfrac{1}{2} |v|^2 + \tfrac{1}{2} h^2 - \tfrac{a}{3} h^3 \, \textup{d}x$ is the Hamiltonian, i.e., the total energy of the system. For $c=2d$, the energy thus decreases

Figures (3)

  • Figure 1: Comparison of simulated enthalpy profiles $h(t)$ for the Westervelt, Kuznetsov and Rasmussen equations. The images correspond to time $t \in \{1, 1.5,2,3\} \cdot 10^{-5}$ (from top left to bottom right).
  • Figure 2: Comparison of the Hamiltonian $\mathcal{H}(v,h)$ for the Westervelt, Kuznetsov and Rasmussen equation in the one-dimensional (left) and two-dimensional example (right).
  • Figure 3: Linear (upper left) vs Westervelt (upper right) vs Kuznetsov (lower left) vs Rasmussen (lower right) for $t \in \{2.5,5,10,20\}\cdot 10^{-5}$.

Theorems & Definitions (14)

  • proposition 1
  • proof
  • remark 1
  • theorem 1: Well-posedness and energy decay
  • proposition 2
  • proof
  • proposition 3
  • proof
  • remark 2
  • proposition 4
  • ...and 4 more